We already observed that as a consequence of Kolmogorov’s continuity theorem, the Brownian paths are -Hölder continuous for every . The next proposition, which is known as the law of iterated logarithm shows in particular that Brownian paths are not -Hölder continuous.

**Theorem.** Let be a Brownian motion. For ,

**Proof**

Thanks to the symmetry and invariance by translation of the Brownian motion, it suffices to show that:

Let us first prove that

Let us denote

Let , from Doob’s maximal inequality applied to the martingale , we have for :

Let now . Using the previous inequality for every with

yields when ,

Therefore from Borel-Cantelli lemma, for almost every , we may find such that for ,

But,

implies that for ,

We conclude:

Letting now and yields

Let us now prove that

Let . For , we denote

Let us prove that

The basic inequality

implies

with

When ,

therefore,

As a consequence of the independence of the Brownian increments and of Borel-Cantelli lemma, the event

will occur almost surely for infinitely many ‘s. But, thanks to the first part of the proof, for almost every , we may find such that for ,

Thus, almost surely, the event

will occur for infinitely many ‘s. This implies

We finally get

by letting

As a straightforward consequence, we may observe that the time inversion invariance property of Brownian motion implies:

**Corollary.** Let be a standard Brownian motion.

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